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Let's imagine a generator that gives you one of the numbers from $[1; n]$, each with probability $1/n$, whenever you push a button. What is the mean amount of button pushes you need to get all numbers from $1$ to $n$?

I've considered using a Markov chain and computing its absorption time, I couldn't compute the fundamental matrix for this chain.

lsparki
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1 Answers1

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Hint: If you have already seen $k$ of the $n$ numbers, the probability that the next number is new is $1-\frac kn$, hence the expected number of button pushes to learn the next number is $\frac1{1-\frac kn}$.

Hagen von Eitzen
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